International Workshopon

Fluid-Structure Interaction Problems

Prague

30 October - 2 November 2007

Place: Blue Lecture Hall, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, Prague

Date: 30 October - 2 November 2007Internet: http://www.karlin.mff.cuni.cz/~prusv/fsi

ProgramTuesday, 30. 10 .2007

09:15-10:00Jiří NeustupaExistence of Weak Solution to the Navier-Stokes Equation in a General Time-Varying Domain by the Rothe Method

10:00-10:45Yoshihiro ShibataSome Stability of Incompressible Viscous Flow Past a Rotating Body

10:45-11:15 coffee break

11:15-12:00Toshiaki HishidaThe Navier-Stokes Flow Around a Rotating Obstacle with Time-Dependent Body Force

12:00-14:00 lunch break

14:00-14:45Matthieu HillairetCollisions Between Rigid Bodies in an Incompressible Viscous Fluid

14:45-15:30Irina DenisovaGlobal Solvability of an Interface Problem Governing the Motion of Two Incompressible Fluids

15:30-16:00 coffee break

16:00-16:45Okihiro SawadaSolvability and Regularity of Solutions to the Navier-Stokes Equations with Unbounded Initial Data

16:45-17:30Elisabetta RoccaA New Dual Approach for a Class of Phase Transitions with Memory

Chairman: Eduard Feireisl (morning)Yoshihiro Shibata (afternoon)

Wednesday, 31. 10. 2007

09:15-10:00Robert DenkBoundary Value Problems with Inhomogeneous Symbols

10:00-10:45Agnieszka Świerczewska–GwiazdaMonotonicity Methods in Generalized Orlicz Spaces for Non-Newtonian Flows

10:45-11:15 coffee break

11:15-12:00Šárka NečasováOn the Motion of Several Rigid Bodies in an Incompressible Non-Newtonian Fluid

12:00-14:00 lunch break

14:00-14:45Ana SilvestreOn the Existence of Time-Periodic Motions of a Rigid Body in a Navier-Stokes Liquid

14:45-15:30Frederic WellerPlatelet Adhesion and Thrombus Growth—A Mathematical and Computational Investigation

15:30-16:00 coffee break

16:00-16:45Stanislav KračmarAnisotropic L 2 Estimates of Weak Solutions to the Stationary Oseen Type Equations in 3D Exterior Domains for a Rotating Body

Chairman: Toshiaki Hishida (morning)Jiří Neustupa (afternoon)

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Thursday, 1. 11. 2007

09:15-10:00Matthias HieberStability of Ekman Boundary Layers

10:00-10:45Katrin SchumacherStrong Solutions to the Stokes-Equations of a Flow Around a Rotating Body in Weighted Functions Spaces

10:45-11:15 coffee break

11:15-12:00Jaroslav HronNumerical Simulation of Fluid-Structure Interaction in ALE Formulation

12:00-14:00 lunch break

14:00-14:45Miroslav KrbecOn the L q Approach to the Problem of Motion of Fluid Around Rotating Body

14:45-15:30Piotr GwiazdaFlat Metric and Structural Stability of a Nonlinear Population Model

15:30-16:00 coffee break

16:00-16:45Petr PausNumerical Solution of Parametric Mean Curvature Flow

18:00 conference dinner

Chairman: Marius Tucsnak (morning)Matthias Hieber (afternoon)

Friday, 2. 11. 2007

09:15-10:00Michal BenešMoving Boundaries in Material Science

10:00-10:45Maria Lukáčová-MedviďováMathematical and Numerical Modelling of Non-Newtonian Fluids in Compliant Vessels

10:45-11:15 coffee break

11:15-12:00Miloslav FeistauerNumerical Simulation of Flow Induced Vibrations

12:00-12:45Tomáš BodnárOn the Shear-Thinning Extension of Oldroyd Type Viscoelastic Models

12:45-13:30Milan PokornýSteady Compressible Navier–Stokes–Fourier System

13:30 closing of the conference

Chairman: Šárka Nečasová (morning)

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Lectures

Michal BenešCzech Technical University, Prague, Czech Republic

Moving Boundaries in Material ScienceThe contribution will discuss several aspects of mathematical modelling and numerical simulation related to the

microstructure formation in solidification of crystalline materials, and dislocation dynamics in the crystalline lattice. Formulation of mathematical models as well as their numerical solution will be presented.

Scheduled on: 2.11.2007, 09:15-10:00

Tomáš BodnárCzech Technical University, Prague, Czech Republic

On the Shear-Thinning Extension of Oldroyd Type Viscoelastic ModelsIt is well known that the classical Oldroyd-A and B models describing the behavior of viscoelastic liquids do not

exhibit the shear-thinning property for the simple shear flow. On the other hand there is a wide class of viscoelastic shear-thinning materials with significant practical applications. One of the most important representants of this class of liquids is blood. Thus it is of essential importance to extend the range of applicability of Oldroyd type models for shear-thinning fluids. This generalization could be provided by implementation of suitable shear-dependent viscosity into the classical Oldroyd type model. Another approach leeds to so called Johnson-Segalman model which uses the choice of advected derivative to modify the simple shear viscosity in an appropriate way. Both approaches will be presented and discussed with respect to possible applications to blood flow simulations.

Scheduled on: 2.11.2007, 12:00-12:45

Irina DenisovaInstitute of Problems of Mechanical Engineering, St. Petersburg, Russia

Global Solvability of an Interface Problem Governing the Motion of Two Incompressible Fluids

We consider unsteady motion of a drop in another incompressible fluid that are contained in a bounded domain. On the unknown interface between the liquids, the surface tension is neglected. This drop motion is governed by an interface problem for the Navier-Stokes system. First, a local existence theorem is established for the problem in Hölder classes of functions. The proof is based on the solvability of a model problem for the Stokes system with a plane interface which was obtained earlier. Next, for a small initial velocity vector field and small mass forces, we prove global solution existence for the nonlinear problem.

Scheduled on: 30.10.2007, 14:45-15:30

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Robert DenkUniversity of Konstanz, Konstanz, Germany

Boundary Value Problems with Inhomogeneous SymbolsClassical parabolic theory of boundary value problems is based on the Fourier transform and the symbols of

differential operators. Here the homogeneity of the symbols leads to solvability results and uniform a prioriestimates in corresponding Sobolev spaces. In several applications in Mathematical Physics, however, there exists no homogeneous symbol of the operator, and classical parabolic theory does not apply. An example for this is the Stefan problem, a free boundary value problem. It is possible to develop a concept of parabolic boundary problems which includes these examples and gives solvability results and a priori estimates for them. This concept is based on the so-called Newton polygon related to the boundary value problem and the Lopatinskii matrix.

Scheduled on: 31.10.2007, 09:15-10:00

Miloslav FeistauerCharles University, Prague, Czech Republic

Numerical Simulation of Flow Induced VibrationsThe subject of the lecture is the numerical simulation of the interaction of two-dimensional incompressible viscous

flow and a vibrating airfoil. A solid airfoil with two degrees of freedom, which can rotate around the elastic axis and oscillate in the vertical direction, is considered. The numerical simulation consists of the finite element solution of the Navier-Stokes equations coupled with the system of ordinary differential equations describing the airfoil motion. High Reynolds numbers considered (105-106) require the application of a suitable stabilization of the finite element discretization. The time dependent computational domain and a moving grid are taken into account with the aid of the Arbitrary Lagrangian-Eulerian (ALE) formulation of the Navier-Stokes equations. A special attention is paid to the time discretization and the solution of the nonlinear discrete problem on each time level is performed. As a result a sufficiently accurate and robust method is developed, which is applied to the case of flow induced airfoil vibrations with large amplitudes after loosing the aeroelastic stability. The computational results are compared with known aerodynamical data and with results of aeroelastic calculations obtained by NASTRAN code for a linear approximation.

The presented results were obtained in cooperation with J. Horáček from the Institute of Thermomechanics of the Academy of Sciences of the Czech Republic and P. Sváček from the Faculty of Mechanical Engineering of the Czech Technical University.

Acknowledgment: This research is supported under the Grant No. 201/05/0005 of the Czech Grant Agency and the Research Project No. MSM 0021620839 of the Ministry of Education of the Czech Republic.

Scheduled on: 2.11.2007, 11:15-12:00

Piotr GwiazdaUniversity of Warsaw, Warsaw, Poland

Flat Metric and Structural Stability of a Nonlinear Population ModelModels describing time evolution of physiologically structured populations have been extensively studied for many

years, [2], [3]. Traditionally the dynamics of such populations are described by partial differential equations (PDE) of transport type.

The first general results on global existence and stability of the solutions of structured population models were established for the states defined in Banach space L1 [2]. In this case it was possible to prove strong continuity and structural stability of the solutions. However, it is often necessary to describe populations, where all individuals have the same state or their distribution is concentrated in respect to the structure, ie. initial distribution of the individuals is not uniformly continuous with respect to the Lebesgue measure. In such cases it is relevant to consider initial data in the

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space of Radon measures as proposed in [4]. Analytical results on the existence of the solutions are given in [4]. However, continuous dependence of solutions on time and state is shown only in the weak topology.

The above motivated us to study the problem of structural stability of the structured population dynamics. Our approach based on the theory of the nonlinear semigroups in the metric spaces, instead of weak semigroups on Banach spaces. Framework of the Wasserstein metric in spaces of probability measures is a very important issue in the analysis of transport equations and was recently developed, see for example [1]. To apply this framework to the nonlinear structured population models, we modify the Wasserstein metric to the general nonnegative Radon measures. An additional problem comparing to the case considered in [1], is that our system consists of the transport equation with a nonlocal boundary condition.

References:[1] Ambrosio, L., Gigli, N. & Savaré, G. (2005): Gradient flows in metric spaces and in the space of probability measures, Birkhäuser, ETH Lecture Notes in Mathematics[2] Webb G.F. (1985): Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York.[3] Thieme, H. R. Mathematics in population biology. Woodstock Princeton university press. Princeton (2003).[4] Diekmann O. and Getto P. (2005): Boundedness, global existenceand continuous dependence for nonlinear dynamical systems describing physiologically structured populations. J. Differ. Equations, 215, pp. 268-319.

Scheduled on: 1.11.2007, 14:45-15:30

Matthias HieberUniveristy of Darmstadt, Darmstadt, Germany

Stability of Ekman Boundary LayersScheduled on: 1.11.2007, 09:15-10:00

Matthieu HillairetUniversité de Lyon, Lyon, France

Collisions Between Rigid Bodies in an Incompressible Viscous FluidRecent results show that a rigid disk surrounded by an incompressible viscous fluid in a bounded cavity cannot reach

the boundary of the cavity in finite time. In my talk, I shall present the key ingredients in the proofs of these results and discuss some possible extensions.

Scheduled on: 30.10.2007, 14:00-14:45

Toshiaki HishidaNiigata University, Niigata, Japan

The Navier-Stokes Flow Around a Rotating Obstacle with Time-Dependent Body Force

The Navier-Stokes flow around a rotating obstacle with time-dependent body force Abstract: Consider the motion of a viscous incompressible fluid filling the whole space exterior to a rigid body, that is rotating with constant angular velocity, under the action of time-dependent external force. We first discuss this problem without initial condition and prove that there exists a unique solution when both the external force and angular velocity are small enough. Our result implies, as a special case, the existence of time-periodic motion. We next show the stability of the obtained solution with respect to small initial disturbance.

Scheduled on: 30.10.2007, 11:15-12:00

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Jaroslav HronCharles University, Prague, Czech Republic

Numerical Simulation of Fluid-Structure Interaction in ALE FormulationIn this contribution we investigate a monolithic algorithm to solve the problem of time dependent interaction between

an incompressible, possibly non-newtonian, viscous fluid and an elastic solid. The continuous formulation of the problem and its discretization is done in a monolithic way, treating the problem as one continuum and discretized by the finite elements method. The resulting set of nonlinear algebraic system of equations is solved by an approximate Newton method with coupled geometric multigrid linear solver for solving the linear subproblems. We discuss possible efficient strategies of setting up the resulting system and its solution.

Joint work with Martin Mádlik (Charles University, Prague).

Scheduled on: 1.11.2007, 11:15-12:00

Stanislav KračmarCzech Technical University, Prague, Czech Republic

Anisotropic L2 Estimates of Weak Solutions to the Stationary Oseen Type Equations in 3D Exterior Domains for a Rotating Body

We study the Oseen problem with rotational effect in exterior threedimensional domains. Using a variational approach we prove existence and uniqueness theorems in anisotropically weighted Sobolev spaces in the whole three-dimensional space. As the main tool we derive and apply an inequality of the Friedrichs-Poincaré type. For the extension of e results to the case of exterior domains we use a localization procedure.

Joint work with Š. Nečasová and P. Penel.

Scheduled on: 31.10.2007, 16:00-16:45

Miroslav KrbecMathematical Institute, Czech Academy of Sciences, Prague, Czech Republic

On the Lq Approach to the Problem of Motion of Fluid Around Rotating Body

This talk will concern the functional analysis background used recently to prove a priori Lq estimates for solutions to equation describing the Stokes and/or Oseen flow around a rotating body in R2 or R3. After a timedependent change of coordinates the problem is reduced to a stationary Stokes and/or Oseen equation, which allows for employing the technique of the maximal functions, weighted Littlewood-Paley decompositions, Muckenhoupt weights, and their factorization, including one-sided variants of the maximal function and anisotropic clones of Muckenhoupt weights.

Basic references: 1. R. Farwig, M. Krbec and Š. Nečasová: A weighted Lq-approach to Stokes flow around a rotating body. To appear in Ann. Univ. Ferrara, Nuova Ser.2. R. Farwig, M. Krbec and Š. Nečasová: A weighted Lq-approach to Oseen flow around a rotating body. To appear in Math. Models Methods Appl. Sci.

Scheduled on: 1.11.2007, 14:00-14:45

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Maria Lukáčová-MedviďováUniversity of Hamburg, Hamburg, Germany

Mathematical and Numerical Modelling of Non-Newtonian Fluids in Compliant Vessels

We present recent results on mathematical modelling and numerical simulation of blood flow in compliant vessels. Blood is modelled as a complex non-Newtonian fluid with shear-thinning properties. In particular, the Carreu and the Yeleswerapu-Rajagopal models are used to describe nonlinear behaviour of viscosity. In order to describe elastic properties of blood vessels the generalized string model has been used. In the talk we present theoretical results on existence and uniqueness of the weak solution in a domain with given deformation. We also illustrate the model properties on a set of numerical experiments describing blood flow in pathological vessels with stenosis.

The present results has been obtained in cooperation with Dr. A. Zauskova (Hamburg University of Technology, Germany).

Scheduled on: 2.11.2007, 10:00-10:45

Jiří NeustupaMathematical Institute, Czech Academy of Sciences, Prague, Czech Republic

Existence of Weak Solution to the Navier-Stokes Equation in a General Time-Varying Domain by the Rothe Method

We assume that Ωt is a general 3D domain continuously varying in dependence on time. We prove the existence of a weak solution to the NavierStokes equation in {(x, t); 0 < t < T, x Ω∈ t } and its weak continuity in a certain sense. We impose no conditions on the smoothness of t and we also discuss application to flows around striking bodies.

Scheduled on: 30.10.2007, 09:15-10:00

Šárka NečasováMathematical Institute, Czech Academy of Sciences, Prague, Czech Republic

On the Motion of Several Rigid Bodies in an Incompressible Non-Newtonian Fluid

We deal with the problem of several rigids bodies in an incompressible non-Newtonian fluids. From recent article of Starovoitov is known under what conditions the body can not touch, but there were not known the existence of solutions. Our aim is to prove the existence of solution in case when the collisions cannot occurs. We investigate the motion of rigid bodies in non-Newtonian fluids of power-law type and the main difficulty is to prove the compactness of the velocity gradients. Our tool to prove this main difficulty is the harmonic pressure which was introduce by Wolf.

Joint work with Eduard Feireisl and Matthieu Hillairet.

Scheduled on: 31.10.2007, 11:15-12:00

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Petr PausCzech Technical University, Prague, Czech Republic

Numerical Solution of Parametric Mean Curvature FlowThe simulation of the evolution of a family of closed and open curves driven by the normal velocity is studied and the

successful numerical results are shown. Several computational examples are shown, which demonstrate numerical approach to the parametric description of the evolution law.

Scheduled on: 1.11.2007, 16:00-16:45

Milan PokornýCharles University, Prague, Czech Republic

Steady Compressible Navier–Stokes–Fourier System

We study steady flow of a compressible heat conducting fluid in a bounded domain Ω R⊂ 3. We consider the slip boundary condition for the velocity and so-called Newton's boundary condition for the temperature. For the pressure law p(ρ, θ) ρ∼ γ + θρ with γ>3 we show that under reasonable technical assumptions on the data of the problem, there is a weak solution to the above mentioned system such that the density ρ L∈ ∞(Ω), the velocity v W∈ 1,q(Ω) and the temperature θ W∈ 1,q(Ω) for any 1 ≤ q < ∞.

Joint work with Piotr B. Mucha (University of Warsaw).

Scheduled on: 2.11.2007, 12:45-13:30

Elisabetta RoccaUniversity of Milano, Milano, Italy

A New Dual Approach for a Class of Phase Transitions with MemoryThe talk regards a joint work with Michel Frémond and Elena Bonetti, e in which we describe a class of phase

transitions with thermal memory using a dual approach with respect to the energy functionals. More precisely, we use as state variables the phase parameter, the entropy (in place of the absolute temperature), and the history contribution of the entropy flux. The equations are recovered from a generalization of the principle of virtual power (to describe the evolution of the phases), including the effects of micro-motions responsible for the phase transition, and a rescalation of the internal energy balance (to describe the evolution of the entropy). Hence, we prove existence of a solution (in a proper functional framework) for the resulting nonlinear integrodifferential PDE system. Finally, we discuss the long-time behaviour of solutions holding on (0, +∞) characterizing the ω-limit of trajectories.

Scheduled on: 30.10.2007, 16:45-17:30

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Okihiro SawadaKyoto University, Kyoto, Japan

Solvability and Regularity of Solutions to the Navier-Stokes Equations with Unbounded Initial Data

We consider the Cauchy problem of the Navier-Stokes equations with an inital velocity given by -f + u0, where f is a

globally Lipschitz continuous function growing-up at space infinity; u0 is a disturbance of p-th integrable. Using the

Ornstein-Uhlenbek semigroup theory, we obtain the time-local existence and uniqueness of mild solutions. For f(x)=Mx with some matrix M the mild solution is a classical solution with suitable pressure term. Assuming M is skew-symmetric additionally, the solution is analytic in x, nevertheless the semigroup is not analytic.

Scheduled on: 30.10.2007, 16:00-16:45

Katrin Schumacher

Strong Solutions to the Stokes-Equations of a Flow Around a Rotating Body in Weighted Functions Spaces

We consider the motion of a fluid in the exterior of a rotating obstacle and assume that this flow is modelled by the Navier-Stokes equations. Then a change of coordinates and a linearization leads to a modified version of the Stokes system which we consider in the whole space Rn, n=2 or n=3 and in an exterior domain D R⊂ 3. The case of weak solutions in exterior domains has been treated by Hishida [2]. In the strong context treated here for every q (1, ∞)∈ existence and estimates are shown in weighted homogeneous Sobolev spaces. The weight functions are taken from the same subclass of the Muckenhoupt class Aq as in [1]. The solutions are unique modulo a vector space of dimension 3.

Joint work with Šárka Nečasová.

References:[1] Farwig, R., Krbec, M. and Necasova, S., A weighted Lq approach to Stokes flow around a rotating body. To appear in Ann. Univ. Ferrara - Sez. VII.[2] Hishida, T., Lq estimates of weak solutions to the stationary Stokes equations around a rotating body. J. Math. Soc. Japan 58 (2006), 743--767.

Scheduled on: 1.11.2007, 10:00-10:45

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Yoshihiro ShibataWaseda University, Tokyo, Japan

Some Stability of Incompressible Viscous Flow Past a Rotating Body

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Scheduled on: 30.10.2007, 10:00-10:45

Ana SilvestreInstituto Superior Tecnico, Lisbon, Portugal

On the Existence of Time-Periodic Motions of a Rigid Body in a Navier-Stokes Liquid

In this talk, we consider a mechanical system {S, L} constituted by a rigid body S moving in a Navier-Stokes liquid L that fills the three dimensional region exterior to S.

The motion of the system {S, L} is described in a reference frame attached to the body and the external forces and torques acting on S and L are given in an inertial frame, so that they become unknown in the frame in which we study the motion of the system.

We present recent results on time-periodic solutions for the fluid-structure interaction problem associated with {S, L}.

This is a joint work with Giovanni Paolo Galdi (University of Pittsburgh, USA).

Scheduled on: 31.10.2007, 14:00-14:45

Frederic WellerUniversity of Heidelberg, Heidelberg, Germany

Platelet Adhesion and Thrombus Growth—A Mathematical and Computational Investigation

Hemostasis is responsible to stem blood loss after injury by platelet plug formation. Although being life essential, a major part of deaths in the western society is due to thrombotic events provoked by disorders of the hemostatic system. Therefore, a better understanding of the underlying mechanisms is needed.

This talk investigates the influence of flow, particularly of shear stress, on platelet adhesion and aggregation. For this purpose, two mathematical models based on the Navier-Stokes equations and on particle conservation are developed: the first one is formulated on a fixed domain and shall capture the initial phase of platelet adhesion, whereas the second model is a free boundary problem that describes long term flow disturbances. Numerical simulations are based on finite elements and the level set method.

Three vessel geometries of physiological relevance are considered: stagnation point flow, sudden expansion and t-junction. Model parameters have been optimized to fit corresponding experimental data. When platelet adhesion is

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assumed independent of shear, numerically predicted spatial platelet distribution does not match these data at all. However, when adhesion is assumed shear-dependent, better agreement is achieved. Regarding the first model, further notable improvement is obtained when surface saturation effects are taken into account. Limitations due to the complexity of the hemostatic system are discussed, as well as possible applications in practice.

The last part of the talk is concerned with the analysis of the free boundary problem and sketches the main steps to proof classical solvability.

Scheduled on: 31.10.2007, 14:45-15:30

Agnieszka Świerczewska–GwiazdaUniversity of Warsaw, Warsaw, Poland

Monotonicity Methods in Generalized Orlicz Spaces for Non-Newtonian FlowsOur interest is directed to non-Newtonian fluids of strongly inhomogeneous behavior with a high ability of increasing

their viscosity under different stimulus, like the shear rate, magnetic or electric field. There are numerous branches of industry, military and natural science, where the application of such fluids is of great importance. Contrary to standard power-law type rheology we propose the formulation with help of space-dependent convex function. This framework includes e.g. rapidly shear thickening and magnetorheological fluids. We provide existence of weak solutions. The nonstandard growth conditions yield the analytical formulation of the problem in generalized Orlicz spaces. The concept of the so-called N-function is introduced, which leads to the definition of the generalized Orlicz space (known also as the Musielak-Orlicz space).

Part of the difficulties in mathematical analysis of such models are caused by the generalized Orlicz spaces themselves. Although they create a natural framework for such analysis, they exhibit new problems arising from the fast growth of an N-function, like the lack of reflexivity of the space or the lack of the density of smooth functions. The literature provides numerous results on the existence of solutions to abstract elliptic and parabolic problems in Orlicz spaces, but the framework of Musielak-Orlicz spaces for non-Newtonian flows is still a developing field.

Scheduled on: 31.10.2007, 10:00-10:45

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